Dynamics of Invasion Percolation

Abstract

Simulations of the growth of planar invasion percolation clusters exhibit novel dynamic scaling. The probability to invade a site at a distance $r$ from a reference site at a time $t$ after that site was invaded behaves as $N(r,t)=r^{-1\mathrm{}}f\left(\frac{r^D}{t}\right)$, where $D$ is the fractal dimension of the invaded region. The scaling function has the unusual timing behavior $f\left(u\right)\mathrm{}\sim u^a(u\ll 1)$ and $\sim u^{-b}\left(u\gg 1\right)$, with $a\simeq 1.4$, $b\simeq 0.6$, and growth occurring mainly around $r\sim t^{\frac{1}{D}}$.